ASTR 1210 (O'Connell) Study Guide

8. GRAVITATIONAL ORBITS AND SPACE FLIGHT

Space Shuttle Discovery launches on a mission
to the Space Station, 2001

"There will certainly be no lack of human pioneers when we have
mastered the art of [space] flight....Let us create vessels and sails adjusted
to the heavenly ether, and there will be plenty of people unafraid of
the empty wastes. In the meantime we shall prepare, for the brave
sky-travelers, maps of the celestial bodies."

---- Johannes Kepler (1610)

Kepler was right about the multitudes of people eager to travel into
space, but it took another 350 years of technological development to
build the "vessels" needed to carry them. Space travel is difficult.
However, the theoretical key to space flight was discovered by
Newton only 80 years after Kepler's work.
Newton's theories of dynamics and gravity provided a complete
understanding of the interaction between gravitating bodies and the
resulting orbits for planets and satellites. This guide first
describes the nature of possible gravitational orbits
and some implications of those.
In the mid-twentieth century, Newton's work became the foundation
of space technology, which is introduced in the second part of
the guide. Space technology---rockets, the Space Shuttle, scores of
robot spacecraft, the human space program---has provided most of our
present knowledge of the Solar System and most of the material we will
discuss in the rest of this course.

A. Newtonian Orbit Theory

Orbital Dynamics

Newton's theory can accurately predict gravitational orbits
because it allows us to determine the acceleration of an object
in a gravitational field. Acceleration is the
rate of change of an object's velocity.

If we know the
initial position and velocity of an object, knowing its acceleration
at all later times is enough to completely determine its later path of
motion.
To predict the path, we simply substitute Newton's expression
for Fgrav for the force term in his Second
Law and solve for acceleration.

But there is a major complication. The Second Law is not a simple
algebraic expression. Both velocity and acceleration are rates of
change (of position and velocity, respectively). Mathematically,
they are derivatives. The gravitational force also changes with
position. Finally, velocity, acceleration and the gravitational force
all have a
directionality as well as a magnitude associated with
them. That is, they are "vectors".
So the Second Law is really a differential vector
equation. To solve it, Newton had to
invent calculus.

We don't need to know the mathematical details in order to
understand the basic interaction that shapes Newtonian orbits. Take
as an example the orbit of the Earth around the Sun.

Pick any location on the Earth's orbit. Represent its velocity at that
location as an arrow (a vector) showing the direction and magnitude
of its motion.
An essential element of Newtonian theory is that changes in
the magnitude of the velocity vector (the speed) or in
the direction of motion are both considered to be
"accelerations." In the following drawings, the red arrows represent
the Earth's velocity vector and the blue arrows represent the applied
gravitational force. According to Newton's Second Law, the change
in the velocity vector (a speed-up in the first case or a
deflection of the direction of motion in the second) is in the
direction of the applied force.

Starting from any location, the instantaneous velocity vector and the
rate of change of that vector (the acceleration) combine to determine
where the Earth will be at the next moment of time. Adding up the
motion from one moment to the next traces out the orbital path.
In Newtonian gravity, the gravitational force acts
radially --- i.e. along the line connecting the Earth and the
Sun. Accordingly, both the
acceleration and the change in the Earth's velocity
vector from one moment in time to the next will also always be in
the radial direction.
You might think that if the acceleration is always toward the Sun,
then the Earth should fall faster and faster on a radial trajectory
until it crashes into the Sun. That's exactly what would
happen if the Earth were ever stationary in its orbit.
In that case, the situation in the left hand drawing above (straight-line
acceleration toward the Sun) would prevail.
But if the Earth's velocity vector has a component which is perpendicular
to the radial direction, then in any interval in time, it will move
"sideways" at the same time as it accelerates toward the Sun.
If the combination of sideways motion and distance from the Sun is
correct, the Earth will avoid collision with the Sun, and it will
stay in permanent orbit. The animation at the right shows the
situation for the Earth's (exaggerated) elliptical orbit around the
Sun (here, the blue line is the velocity vector, the green line is the
acceleration;
click for an
enlargement). Note that where
the Earth is nearest the Sun, the gravitational force and inward acceleration
are greatest, but the sideways motion is also greatest, which prevents us
from colliding with the Sun. That motion is in accordance with
Kepler's
second law.
Therefore, all permanently orbiting bodies are perpetually
falling toward the source of gravity but have enough sideways
motion to avoid a collision.

Kinds of Gravitational Orbits

In the case of two gravitating objects (for example, the Earth and the
Moon, the Sun and a planet, or the Earth and an artificial satellite),
Newton found that the full solutions of his equations give the
following results:

The relative orbit is confined to a geometric plane which
contains both objects.

The shape of the orbit within the plane is a "conic section",
of which there are only four types.

A circle

An ellipse

A parabola

A hyperbola
See the illustration at the right.

The orbital type is determined by the
initial distance, speed and
direction of motion of the orbiting object, as follows:

Define the "escape velocity" at a given
distance: Vesc(R) = √(2GM/R), where R
is the separation between the two objects and M is
the mass of the primary object.

Vesc for the Earth at the Earth's surface is 25,000 mph (or 11 km/s).
Vesc for the Sun at the Earth's 1 AU distance from the Sun
is 94,000 mph (42 km/s).

If V < Vesc, the orbit is an ellipse
or circle. It is said to be "closed" or
"bound". The smaller object will permanently
repeat its orbital motion.

If V ≥ Vesc, the orbit is a parabola or
hyperbola. It is said to be "open" or "unbound".
The smaller object escapes and does not return.

Only specific values of velocity will yield circular or
parabolic
orbits. An object moving exactly at escape velocity will move on a
parabola. To achieve a circular orbit an object must move at 71% of
the escape velocity, and its velocity must be exactly
perpendicular to the radial direction.

As noted earlier, shapes and motions within the "closed" orbits
for the planets satisfy all three of Kepler's Laws of planetary
motion.

You can interactively explore the relation between the orbit and the
initial velocity vector using the Flash animation
Gravity Chaos.

Newton's Mountain

Newton illustrated orbital behavior for a simple idealized situation
where a powerful cannon is fixed in position on top of a high mountain
on the Earth's equator. It is allowed to fire only with its barrel
parallel to the Earth's surface (see the illustration below). Since
both the distance from Earth's center and the direction of initial
flight are fixed, the cannonball follows an orbit that depends only
on the muzzle velocity of the cannon as shown below.

The gravitational force of a spherical body like the Earth acts as
though it originates from the center of the sphere, so
elliptical orbits of the cannonball will have the center of the Earth
at one focus. The center of the Earth lies directly under the red
dot in the picture below.
(For simplicity, the diagram omits showing "orbits" where the muzzle
velocity is far enough below the circular orbit velocity that the
cannonball hits the surface of the Earth. Needless to say, that is
the situation for all real-life cannons; but the flight paths taken
before the collision do have elliptical shapes.)

General Relativity

Much later (1915), Newton's theory was shown to be inadequate by
Albert Einstein in the presence of large masses or over large
distances and has been replaced by the General Theory of Relativity
in such situations. Relativity theory profoundly changed our
understanding of space and time, for example by demonstrating that
mass and energy can affect the structure of space and time. It is
much more complicated mathematically than Newton's formulation. But
as a practical matter, Newton's theory is an entirely satisfactory
description of "everyday" gravity. Only very minor corrections to the
Newtonian predictions are necessary, for example, to send spacecraft
with high accuracy throughout the solar system.

General Relativity predicted important phenomena that did not exist in
Newtonian physics,
including black
holes (closed regions of space-time around massive, compact
objects)
and
gravitational waves (propagating disturbances in space-time).
The first detection of cosmic gravitational waves, generated by the
merger of two black holes, was announced by
LIGO (the Laser
Interferometer Gravitational-Wave Observatory) in February of 2016.
This was one of the most difficult scientific experiments ever
attempted.

B. Important Implications of Newtonian Orbits

"Free-Fall" Orbits

Free motion in response to gravity (in the absence of other forces) is
called "free-fall" motion. Conic section
orbits are all "free-fall orbits."

Remember that motion is normal in free-fall. For instance,
engines do not have to be on in order for spacecraft to
move through space on a free-fall orbit. Spacecraft will "coast" forever on
such an orbit, just as do the planets in orbit around the Sun.
Note also that free-fall orbits will depart from simple conic
sections if an object is under the influence of
more than one gravitating body. For instance, comets
are often deflected from their Sun-dominated simple conic orbits by
Jupiter's gravity (see Guide 21),
and spacecraft traveling between the Earth and the Moon will not
follow simple conic paths.

Free-fall orbits are independent of the mass of the orbiting object.

The mass of the orbiting body always
cancels out of the expression for acceleration under gravity.
For instance, in the case of a planet orbiting the Sun, the
gravitational force on the planet is directly proportional to
the planet's mass; but, according to Newton's Second Law, the resulting
acceleration is inversely proportional to its mass. Hence,
mass drops out of the expression for acceleration.
This is true for all orbits under gravity. Hence, a tennis ball in
space, if it were moving with the same speed and direction as the
Earth at any point, would follow exactly the same orbital path as the
Earth.
Kepler's Third Law (that the orbital period of a planet around the Sun
depends only on orbital size, not on the mass of the planet) is another
manifestation of this fact.

The characteristics of free fall originate from the fact that
the acceleration of all objects is the same in a given gravity
field (e.g. at a given distance from the Sun or near the Earth's
surface), regardless of their masses. This was first demonstrated
experimentally
by Galileo
and was the subject of a Puzzlah (see also Study Guide 7)
A more familiar manifestation these days is the phenomenon
of "floating" astronauts on space missions. Even though the
spacecraft is much more massive, both the astronauts and the
spacecraft have identical accelerations under the external
gravitational fields. They are moving on parallel free-fall
orbits, so the astronauts appear to be floating and stationary with
respect to the spacecraft, even though in near-Earth orbit they are
actually both moving at tens of thousands of miles per hour.

Rocket engines are described under (C) below. You can think of a
rocket engine in the abstract as a device for
changing from one free-fall orbit to another by applying a
non-gravitational force.

With its engine turned off, the motion of any
spacecraft is a free-fall orbit.
If the engine is on, the craft is
not in free fall. For instance, the orbit of the Space Shuttle
launching from the Earth will depart from a conic section until its
engines turn off. An example of using a rocket engine to change from
one free-fall orbit to another
is shown here.

The Russian "Mir" space station (1986-2001) orbiting
Earth at an altitude of 200 miles with a velocity of 17,000
mph

Geosynchronous Orbits

According to Kepler's Third Law, the
orbital period of a satellite will
increase as its orbital size increases. We have exploited that
fact in developing one of the most important practical
applications of space technology: geosynchronous satellites.

Spacecraft in "low" Earth orbits (less than about 500 mi),
like the Mir space station (seen above) or the Space Shuttle, all
orbit Earth in about 90 minutes, at 17,000 miles per hour,
regardless of their mass.

The orbital period of a spacecraft in a
larger orbit will be longer. For an orbit of radius
about 26,000 mi, the period will be 24 hours---the same
as the rotation period of the Earth. Spacecraft here, if they are
moving in the right direction, will appear to "hover" over a
given point on the Earth's surface. These orbits are therefore called
geosynchronous or "geostationary." See the animation
above. This is the ideal location for placing communications satellites.
[The concept of geosynchronous communications satellites was first
proposed by science fiction
writer Arthur
C. Clarke. He deliberately did not patent his idea, which became
the basis of a trillion-dollar industry.]

Applications of Kepler's Third Law

Newton's theory verified Kepler's
Third Law (described in Guide 7) and provided a physical
interpretation of it. Newton found, as did Kepler, that P2
= K a3, where P is the period of a planet in its
orbit, a is the semi-major axis of the orbit, and K is a
constant. But his gravitational theory allowed him to show that the
value of K is K = 4π2/GM, where M is the
mass of the Sun.
More generally, K is inversely proportional to the mass of the
primary body (i.e. the Sun in the case of the planetary orbits but
the Earth in the case of orbiting spacecraft). The larger the
mass of the primary, the shorter the period for a given
orbital size.

The Third Law therefore has an invaluable astrophysical
application: once the value of the "G" constant has been determined
(in the laboratory), the motions of orbiting objects can be used to
determine the mass of the primary. This is true no matter how far
from us the objects are (as long as the orbital motion and size can be
measured).

In the Solar System, the Third Law allows us to determine the
mass of the Sun from the size and periods of the planetary orbits.
It allows us to determine the mass of other planets; in the case of
Jupiter, for example, the periods and sizes of the orbits of the
Galilean satellites can be used to determine Jupiter's mass (as
in Optional Lab 3).

The Third Law was critical in determining the masses of other
stars, using their orbits in binary star systems, and hence to
deducing the physical processes that
control
stellar structure and evolution, one of the great accomplishments
of 20th-century physics.

The Third Law is applied today to such diverse astronomical
problems as measuring the masses of "exoplanets" around other
stars (see Study Guide 11) and establishing
the existence of "Dark Matter" in distant galaxies.

Schematic diagram of a liquid-fueled rocket engine.
Rockets carry both fuel and an oxidizer, which allows the fuel to burn
even in the absence of an oxygen-rich atmosphere. The thrust of the
engine is proportional to the velocity of the exhaust gases
(Ve).

C. Space Flight

If the primary technology enabling space flight is Newtonian
orbit theory, the second most important technology is the
rocket engine.

In a rocket engine such as that shown in the diagram above, fuel
is burned rapidly in a combustion chamber and converted into a large
quantity of hot gas. The gas creates high pressure, which causes it
to be expelled out a nozzle at very high velocity.

The exhaust pressure
simultaneously forces the body of the rocket forward. You can think
of the rocket as "pushing off" from the moving molecules of exhaust
gas. The higher the exhaust velocity, the higher the thrust.

Note: rockets do not "push off" against the air or against the
Earth's surface. Rather, it is the "reaction force" between the
expelled exhaust and the rocket that impells the rocket forward.

Designers work to achieve the highest possible exhaust velocity per
gram of fuel. Newton's second law of motion and various elaborations
of it are essential for understanding and designing rocket motors.

The main challenge to spaceflight is obtaining the power needed to
reach escape velocity. For Earth, this is 11 km/sec or 25,000 mph.

"Standard" rocket engines are designed for launching commercial
payloads to synchronous orbit or delivering intercontinental ballistic
missles---neither of which involve reaching escape velocity from
Earth. Therefore, most scientific spacecraft for planetary missions
are relatively small (i.e. low mass) in order that standard engines
can propel them past Earth escape velocity. This means that many
clever strategies are needed to pack high performance into small,
light packages.

Example: The New Horizons
spacecraft, launched on a super-high velocity trajectory to Pluto in
2006, has a mass of only 1050 lbs; its launching rocket weighed
1,260,000 lbs, over 1000 times more! New Horizons flew past
Pluto in July 2015, traveling at over 34,000 mph, and has delivered a
treasure-trove of images and other data on this most distant (3
billion miles from Earth at the encounter) of the classical
planets.
A rocket launched at exactly escape velocity from a given parent
body will, at very large distances, slow to exactly zero velocity
with respect to that body (ignoring the effect of other gravitating
bodies).

The Apollo program used the extremely
powerful Saturn V rockets to
launch payloads with masses up to 100,000 pounds (including 3 crew
members) to the Moon. This technology was, however, retired in the
mid-1970's because it was thought, erroneously, that the next
generation of "reusable" Space Shuttle vehicles would be cheaper to
operate. A big mistake, because the reusable parts required
extensive (and expensive) refurbishing for each new flight.
The Space Shuttle (shown above) was fueled by high energy
liquid oxygen and liquid hydrogen plus solid-rocket boosters. But it
was so massive compared to the power of its engines that it
could not reach escape velocity from Earth. Its maximum altitude
is only about 300 miles. That is why NASA and the private sector are
developing new
"heavy lift" rocket technologies to replace the Shuttle.

Modern rocket engines are remarkably
complicated. Here
is a simplified schematic of the Space Shuttle main engines.

D. Interplanetary Space Missions

The mid-20th century was the first time humans had ever sent
machines beyond the Earth's atmosphere. By 2015, we had explored
every large body in the Solar System out to the orbit of
Pluto. Even such far-sighted thinkers as Galileo and Newton would
never have thought that possible in the mere 400 years that had elapsed
since Kepler's Laws were formulated. This was an amazing
accomplishment, the greatest exploratory feat of humanity to date.

Beginning in the early 1960's, NASA and
foreign space agencies developed a series of ever-more sophisticated
robot probes to study the Sun, Moon, planets, and the
interplanetary medium. These included flyby spacecraft,
orbiters, landers, rovers, and sample-return vehicles.
As of 2015, only 58 years after the first successful satellite launch
(Sputnik, in 1957), we had flown at close range past every planet
including Pluto; had placed robotic observatories into orbit around
the Moon, Mercury, Venus, Mars, Jupiter, Saturn, three asteroids, and
one comet; had sent probes into a comet nucleus and the atmosphere of
Jupiter; had soft-landed on the Moon, Venus, Mars, Saturn's moon
Titan, and the comet Churyumov-Gerasimenko; and had returned to Earth
samples obtained from the coma of the comet Wild 2 and from a
soft-landing on the asteroid Itokawa. And in 2013, the first human
spacecraft left the Solar System altogether and entered interstellar
space. At right is an artist's concept painting of the Cassini
mission in orbit around Saturn.

We also put a number of highly capable remote-controlled observatories
for studying the Solar System and the distant universe (such as the
Hubble Space Telescope and the Chandra X-Ray Observatory) into orbit
around the Earth and the Sun.
Of course, the Apollo program in the 1960's also sent human beings
to the Moon. This was a triumph of human courage, skill, and
organization and was very fruitful in learning about lunar geology and
surface history. But, by far, most of what we know about the
denizens of the Solar System has come from our powerful robot
missions and observatories.
For a list of these missions and additional
links, click here.